Showing papers on "Linear map published in 1995"

Ill-posed problems with a priori information

Abstract: Part 1 Unstable problems: base formulations of problems ill-posed problems examples and its stability analysis the classification of methods for unstable problems with a priori information. Part 2 Iterative methods for approximation of fixed points and their application to ill-posed problems: basic classes of mappings convergence theorems for iterative processes iterations with correcting multipliers applications to problems of mathematical programming regularizing properties of iterations iterative processes with averaging iterative regularization of variation inequalities and of operator equations with monotone operators iterative regularization of operator equations in the partially-ordered spaces iterative schemes based on the Gauss-Newton method. Part 3 Regularization methods for symmetric spectral problems: L-basis of linear operator kernel analogies of Tikhonov's and Lavent'ev's methods the variational residual method and the quasisolutions method regularization of generalized spectral problem. Part 4 The finite-moment problem and systems of operators equations: statement of the problem and convergence of finite-dimensional approximations iterative methods on the basis of projections the Fejer processes with correcting multipliers FMP regularization in Hilbert spaces with reproducing kernels iterative approximation of solution of linear operator equation system. Part 5 Discrete approximation of regularizing algorithms: discrete convergence of elements and operators convergence of discrete approximations for Tikhonov's regularizing algorithm applications to integral and operator equations interpolation of discrete approximate solutions by splines discrete approximation of reconstuction of linear operator kernel basis finite-dimensional approximation of regularized algorithms on discontinuous functions classes. Part 6 Numerical applications: iterative algorithms for solving gravimetry problem computing schemes for finite-moment problem methods for experiment data processing in structure investigations of amorphous alloys. Appendix: correction parameters methods for solving integral equations of the first kind.

Logical/linear operators for image curves

Abstract: We propose a language for designing image measurement operators suitable for early vision. We refer to them as logical/linear (L/L) operators, since they unify aspects of linear operator theory and Boolean logic. A family of these operators appropriate for measuring the low-order differential structure of image curves is developed. These L/L operators are derived by decomposing a linear model into logical components to ensure that certain structural preconditions for the existence of an image curve are upheld. Tangential conditions guarantee continuity, while normal conditions select and categorize contrast profiles. The resulting operators allow for coarse measurement of curvilinear differential structure (orientation and curvature) while successfully segregating edge-and line-like features. By thus reducing the incidence of false-positive responses, these operators are a substantial improvement over (thresholded) linear operators which attempt to resolve the same class of features. >

Optimally adaptive transform coding

Abstract: The optimal linear block transform for coding images is well known to be the Karhunen-Loeve transformation (KLT). However, the assumption of stationarity in the optimality condition is far from valid for images. Images are composed of regions whose local statistics may vary widely across an image. While the use of adaptation can result in improved performance, there has been little investigation into the optimality of the criterion upon which the adaptation is based. In this paper we propose a new transform coding method in which the adaptation is optimal. The system is modular, consisting of a number of modules corresponding to different classes of the input data. Each module consists of a linear transformation, whose bases are calculated during an initial training period. The appropriate class for a given input vector is determined by the subspace classifier. The performance of the resulting adaptive system is shown to be superior to that of the optimal nonadaptive linear transformation. This method can also be used as a segmentor. The segmentation it performs is independent of variations in illumination. In addition, the resulting class representations are analogous to the arrangement of the directionally sensitive columns in the visual cortex. >

A robust PI-controller for infinite-dimensional systems

Abstract: In this paper, we deal with single-input single-output systems of the form on a separable Hilbert space H, where the operator A is the generator of an exponentially stable C0-semigroup on H, b ϵ H, C is a A -admissible linear operator and w is an arbitrary constant disturbance vector in H. We propose a low-gain PI-controller which stabilizes and regulates the system such that, for a given reference constant yr, y(t) tends to yr independently of w as t → + ∞ . Our result generalizes the previous one of Pohjolainen (1982) in that the semigroup is not necessarily holomorphic. A numerical example will be given to illustrate the application of the theory.

Parametrization of finite rotations in computational dynamics: a review

Abstract: Finite rotations are traditionnally regarded as geometric operations on vectors. By adopting an algebraic point of view, they many also be regarded as linear transformations with invariance propert...

Beyond unimodular transformations

Abstract: This paper presents an approach to modeling loop transformations using linear algebra. Compound transformations are modeled as integer matrices. The nonsingular linear transformations presented here subsume the class of unimodular transformations. The loop transformations included are the unimodular transformations-reversal, skewing, and permutation- and a new transformation, namelystretching. Nonunimodular transformations (with determinant ≥ 1) create “holes” in the transformed iteration space, rendering code generation difficult. We solve this problem by suitably changing the step size of loops in order to “skip” these holes when traversing the transformed iteration space. For the class of nonunimodular loop transformations, we present algorithms for deriving the loop bounds, the array access expressions, and the step sizes of loops in the nest. To derive the step sizes, we compute the Hermite normal form of the transformation matrix; the step sizes are the entries on the diagonal of this matrix. We then use the theory of Hessenberg matrices in the derivation of exact loop bounds for nonunimodular transformations. We illustrate the use of this approach in several problems such as the generation of tile sets and distributed-memory code generation. This approach provides a framework for optimizing programs for a variety of architectures.

Redundancy reduction with information-preserving nonlinear maps

Abstract: The basic idea of linear principal component analysis (PCA) involves decorrelating coordinates by an orthogonal linear transformation. In this paper we generalize this idea to the nonlinear case. Simultaneously we shall drop the usual restriction to Gaussian distributions. The linearity and orthogonality condition of linear PCA is replaced by the condition of volume conservation in order to avoid spurious information generated by the nonlinear transformation. This leads us to another very general class of nonlinear transformations, called symplectic maps. Later, instead of minimizing the correlation, we minimize the redundancy measured at the output coordinates. This generalizes second-order statistics, being only valid for Gaussian output distributions, to higher-order statistics. The proposed paradigm implements Barlow's redundancy-reduction principle for unsupervised feature extraction. The resulting factorial representation of the joint probability distribution presumably facilitates density estimatio...

The Role of Functional Analysis in Global Illumination

Abstract: The problem of global illumination is virtually synonymous with solving the rendering equation. Although a great deal of research has been directed toward Monte Carlo and finite element methods for solving the rendering equation, little is known about the continuous equation beyond the existence and uniqueness of its solution. The continuous problem may be posed in terms of linear operators acting on infinite-dimensional function spaces. Such operators are fundamentally different from their finite-dimensional counterparts, and are properly studied using the methods of functional analysis. This paper summarizes some of the basic concepts of functional analysis and shows how these concepts may be applied to a linear operator formulation of the rendering equation. In particular, operator norms are obtained from thermodynamic principles, and a number of common function spaces are shown to be closed under global illumination. Finally, several fundamental operators that arise in global illumination are shown to be nearly finite-dimensional in that they can be uniformly approximated by matrices.

Magnetic field inhomogeneity correction in MRI using estimated linear magnetic field map

Abstract: A fast and robust method for correcting magnetic resonance image distortion due to field inhomogeneity is applied to spiral k-space scanning. The method includes acquiring a local field map, finding the best fit to a linear map, and using the linear field map to deblur the image distortions due to local frequency variations. The linear field map is determined using a maximum likelihood estimator with weights proportional to the pixel intensity. The method requires little additional computation and is robust in low signal regions and near abrupt field changes. The application of this method is illustrated in conjunction with a multi-slice, T2-weighted, breath-held spiral scan of the liver.

On the Tensor Function Representations of 2nd‐Order and 4th‐Order Tensors. Part I

Abstract: In this paper, we establish the complete representations for isotropic scalar-, 2nd-order tensor- and 4th-order tensor-valued functions of any finite number of 2nd- and 4th-order tensors in both 2- and 3-dimensional spaces, and prove the irreducibility of these representations. The fourth-order tensors under consideration are those regarded as skew-symmetric linear transformations of both symmetric and skew-symmetric 2nd-order tensors in 3-dimensional space, symmetric linear transformations of skew-symmetric 2nd-order tensors in 3-dimensional space, and both symmetric and skew-symmetric linear transformations of both symmetric and skew-symmetric 2nd-order tensors in 2-dimensional space. Complete and irreducible isotropic tensor function representations in 3-dimensional space involving 4th-order tensors as symmetric linear transformations of symmetric 2nd-order tensors are derived in a continued paper (Part II). These representations allow general forms of constitutive laws involving 4th-order tensor agencies (damage tensors, internal variables) of isotropic materials to be developed.

Banach Space Complexes

Abstract: Introduction. I: Preliminaries. 1. Algebraic prerequisites. 2. Algebraic Fredholm pairs. 3. Paraclosed linear transformations. 4. Homogeneous operators. 5. Linear and homogeneous projections and liftings. 6. The gap between two closed subspaces. 7. Linear operators with closed range, and finite extensions. 8. Metric relations and duality. 9. Operators in quotient Banach spaces. 10. References and comments. II: Semi-Fredholm complexes. 1. Semi-Fredholm operators. 2. Semi-Fredholm complexes. 3. Essential complexes. 4. Fredholm pairs. 5. Other continuous invariants. 6. References and comments. III: Related topics. 1. Joint spectra and perturbations. 2. Spectral interpolation and perturbations. 3. Versions of Poincare's and Grothendieck's lemmas. 4. Differentiable families of partial differential operators. 5. References and comments. Subject index. Notations. Bibliography.

Regularized semigroups, existence families and the abstract Cauchy problem

Abstract: For an arbitrary bounded linear operator $C$, on a Banach space, and a closable linear operator $A$, we introduce a $C$-regularized semigroup for $A$. We present equivalences between $A$ having a $C$-regularized semigroup, the corresponding abstract Cauchy problem, well-posedness on a continuously embedded subspace, and (for exponentially bounded $C$-regularized semigroups) the Laplace transform.

Time-recursive computation and real-time parallel architectures: a framework

Abstract: The time-recursive computation has been proven a particularly useful tool in real-time data compression, in transform domain adaptive filtering, and in spectrum analysis. Unlike the FFT-based ones, the time-recursive architectures require only local communication. Also, they are modular and regular, thus they are very appropriate for VLSI implementation and they allow a high degree of parallelism. In this two-part paper, we establish an architectural framework for parallel time-recursive computation. We consider a class of linear operators that consists of the discrete time, time invariant, compactly supported, but otherwise arbitrary kernel functions. We show that the structure of the realization of a given linear operator is dictated by the decomposition of the latter with respect to proper basis functions. An optimal way for carrying out this decomposition is demonstrated. The parametric forms of the basis functions are identified and their properties pertinent to the architecture design are studied. A library of architectural building modules capable of realizing these functions is developed. An analysis of the implementation complexity for the aforementioned modules is conducted. Based on this framework, the time-recursive architecture of a given linear operator can be derived in a systematic routine way.

On Some Generalizations of Batalin-Vilkovisky Algebras

Abstract: We define the concept of higher order differential operators on a general noncommutative, nonassociative superalgebra A, and show that a vertex operator superalgebra has plenty of them, namely modes of vertex operators. A linear operator \Delta is a differential operator of order at most r if an inductively defined (r+1)-form \Phi_{\Delta}^{r+1} is identically zero. When A is supercommutative and associative and \Delta is an odd, square zero, second order differential operator on A, \Phi_{\Delta}^2 defines a "Batalin-Vilkovisky algebra" structure on A. We generalize this notion to any superalgebra with an odd, square zero, second order differential operator, and show that all properties of the classical BV bracket but one continue to hold "on the nose". We also point out connections to Leibniz algebras and the noncommutative homology theory of Loday. Taking the generalization process one step further, we remove all conditions on \Delta to examine the changes in the basic properties of the bracket. We see that a topological chiral algebra with a mild restriction yields a classical BV algebra in the cohomology. Finally, we investigate the quantum BV master equation and relate it to deformations of differential graded algebras.

Stochastic Adaptive Control

Abstract: The objective of this paper is to present some identification problems and adaptive control problems for continuous time linear and nonlinear stochastic systems that are completely or partially observed For continuous time linear of stochastic systems the consistency of a family of least squares estimates of some unknown parameters is verified The unknown parameters appear in the linear transformations of the state and the control An approach to the verification of the consistency associates a family of control problems to the identification problem and the asymptotic behavior of the solutions of a family of algebraic Riccati equations from the control problems implies a persistent excitation property for the identification problem The theorem of locally asymptotically normal experiment is used to test hypotheses about the parameters of a controlled linear stochastic system The tests are formulated for both continuous and sampled observations of the input and the output

An interactive analysis method for multidimensional images using a Hilbert curve

Abstract: To analyze multidimensional images we need a mapping of feature vectors from a multidimensional space to a lower dimensional space. In general, these are performed using linear transformation methods, such as principal component analysis, etc. Linear transformation requires many rotations of data from several points of view because the mapping is not one-to-one. Here, a new interactive method for classifying multispectral images using a Hilbert curve is presented. The Hilbert curve is a one-to-one mapping from N-dimensional space to one-dimensional space and preserves the neighborhood as much as possible. Hilbert curve is a kind of space filling curves, and provides a continuous scan. The merit of the system presented is that the user can extract category clusters without computing any distance in N-dimensional space easily. The method presented here is explained in brief. Clusters are extracted from 1-D data mapped by a Hilbert curve interactively, i.e., a pixel is classified as a category. The user can analyze multidimensional images hierarchically from gross data distribution to fine data distribution. To realize the real time response from the system, data tables storing the addresses and the occurrences of data are used. Here, the address is defined by using the coordinates in N-dimensional space, and a part of mapping which cannot preserve the neighborhood is utilized. In the experiments ex-extracting categories from LANDSAT data, it is confirmed that the user can obtain the real time response from the system after once making the data tables.

ALMOST MULTIPLICATIVE MORPHISMS AND THE CUNTZ ALGEBRA ${\mathcal O}_2$

Abstract: Let X be a compact metrizable space, and let A be a purely infinite simple C*-algbera A satisfying K0(A)=K1(A)=0. We show that an almost multiplicative contractive unital *-preserving linear map from C(X) can be approximated by a homomorphism. As a consequence, we show that if a unital simple C*-algbera , with (finite direct sums), for compact metrizable spaces Xm,n and are even algebras , satisfies K0(A)=K1(A)=0, then . In particular, we show that the tensor product of a simple unital AH-algebra with is isomorphic to .

Effects of random linear transformations on higher-order cyclostationary time-series

Abstract: In many signal processing applications, it is advantageous to exploit the signal selectivity and noise tolerance that are typical of the algorithms based on the spectral line generation that often can be accomplished by using quadratic nonlinear transformations The linear time-variant systems are classified as random and nonrandom in the fraction-of-time probability framework Then, the way in which the higher-order wide-sense cyclostationarity properties of time-series change as they are processed by random or nonrandom linear systems is investigated

Multi-class maximum entropy coder

Abstract: The optimal linear block transform for coding images is known to be the Karhunen-Loeve transform (KLT). However, the assumption of stationarity in the optimality condition is far from valid for images. Images are composed of regions whose local statistics may vary widely across an image. The authors propose a new transform coding method which optimally adapts to such local differences based on an information-theoretic criterion. The new system consists of a number of modules corresponding to different classes of the input data. Each module consists of a single-component, linear transformation, whose basis vector is calculated during an initial training period. The appropriate class for a given input vector is determined by the optimal maximum entropy classifier. The performance of the resulting adaptive network is shown to be superior to that of the optimal nonadaptive linear transformation, both in terms of rate-distortion and computational complexity.

Convergence rates for moment collocation solutions of linear operator equations

Abstract: Given a linear operator equation Kf=g with data g(x i )i=1,…,n, we consider the general moment collocation solution defined as the function f n that minimizes ||Pf n ||2 over a Hilbert space, subject to Kf n (x i )=g(x i )i=1,…,n. Here P is an orthogonal projection with a finite dimensional null space. In the case of P=I, the identity, it is known that if a certain kernel depending on K is continuous, then f n → f 0 , the true solution, as the maximum subinterval width → 0. Moreover, if the kernel satisfies a smoothness condition, then rates of convergence are known. In this paper we extend these results to the case with general P.

Applications of generalized inverses to interval linear programs in hilbert spaces

Abstract: Let H1 and H2 be real Hilbert spaces. Suppose H2 is partially ordered, a, b ∊ H2 c ∊ H1 and A :H1 → H2 is a continuous linear map. We consider the following interval linear program: Maximize subject to a ≤ Ax ≤ b. Conditions under which explicit solutions to the above problem can be found are studied. The solutions are represented in terms of generalized inverses of A. Several examples are given to illustrate the main results.

Regularizing conjugate-direction methods

Abstract: The convergence of regularizing conjugate-direction methods for solving ill-posed linear operator equations in Hilbert spaces is proved. Accuracy estimates of optimum order on the set of sourcewise representable solutions are obtained.